In the last post I discussed the second chapter of the book ‘La Vie Oscillatoire’ which is concerned with glycolytic oscillations. The third chapter is on calcium oscillations, a theme which I have written about more than once in recent posts. Here I will say something about the subject of the fourth chapter, signalling by the cellular slime mould Dictyostelium. I have written some things about Dictyostelium in previous posts. It has fascinated many scientists by its ability to go from a state where the cells are independent to a state which looks like a multicellular organism as a reaction to a scarcity of food. The cells gather by means of chemotaxis. It is usual for mathematical talks about chemotaxis to start with nice pictures portraying the life cycle of Dictyostelium discoideum, the most famous organism of this type. It gave rise to the formulation of the Keller-Segel model which is rather popular among mathematicians. As I have mentioned in previous posts it is not so clear to what extent the Keller-Segel model, as attractive as it is, is really relevant to the life of D. discoideum. In his book Goldbeter seems to share this sceptical point of view while leaving open the possibility that the Keller-Segel model might be a reasonable model for aggregation in a less famous relative of D. discoideum, Dictyostelium minutum.
Returning to D. discoideum, it is a known fact that the cells of this organism signal to their neighbours by producing cAMP (cyclic adenosine monophosphate). This process can be modelled by a system of ODE (or a similar system with diffusion) called the Martiel-Goldbeter model. Experimentally it is seen that cultures of starving D. discoideum develop circular or spiral waves centered on certain pacemaker cells. In the book a decription is given of how this process can be understood on the basis of the Martiel-Goldbeter model. It is useful to draw a diagram of the dynamical properties of this model as a function of the activities of two key enzymes. There is a region where solutions of the MG model converge to a stationary solution, a region where they show excitable behaviour and a region where there is a limit cycle. As a cell develops following the beginning of a period of hunger it moves around in this parameter space. It starts from constant production (1), becomes excitable (2), produces pulsations (3) and then becomes excitable again (4). This is a statement which relates to the ODE system but when diffusion is added the idea is that the pacemaker cells are in stage (3) of their development while cells in stage (4) can then lead to waves which propagate away from the pacemakers. Because of the variability of the cells (of their intrinsic properties or their life histories) the different stages of development can be present at the same time.
I now want to say some more about the MG model itself. The variables in the system are the fraction of receptor in the active state and the intracellular () and extracellular () concentrations of cAMP. The evolution equation for is of the form where and are ratios of linear functions. The evolution equation for is where the quantities other then the unknowns are constants. The most complicated evolution equation is that for which is . Here is the concentration of ATP, taken to be constant in this model, and all other quantities except the unknowns are constants. The function is complicated and will not be written down here. It is obtained from a system with more equations by a quasi-steady state assumption and in its dependence on it is a ratio of two quadratic functions. A further quasi-steady state assumption leads to a simplified system for and alone which is more tractable for analytical considerations.